Buoyancy, Flotation and Stability • When a stationary body is completely submerged in a fluid, or floating (partially submerged), the resultant fluid force on the body is the buoyant force. • A net upward force results because • Buoyant force has a magnitude equal to the weight of the fluid displaced by body and is directed vertically upward. • Archimedes’ principle (287-212 BC)

Buoyant force passes through the centroid of the displaced volume Figure 2.24 (p. 70) Buoyant force on submerged and floating bodies.

Example 1 A spherical buoys has a diameter of 1.5 m, weighs 8.50 kN and is anchored to the seafloor with a cable. What is the tension on the cable when the buoy is completely immersed?

Example 2 Measuring specific gravity by a hydrometer

Stability of Immersed and Floating Bodies • Centers of buoyancy and gravity do not coincide • A small rotation can result in either a restoring or overturning couple. • Stability is important for floating bodies

Stability of an immersed body Stability of a completely immersed body – center of gravity above centroid. Stability of a completely immersed body – center of gravity below entroid.

Stability of a floating body

Elementary Fluid Dynamics • Newton’s second law • Bernoulli equation (most used and the most abused equation in fluid mechanics) • Inviscid flow- flow where viscosity is assumed to be zero; viscous effects are relatively small compared with other effects such as gravity and pressure differences. • Net pressure force on a particle +net gravity force in particle • Two dimensional flow (in x-z plane) • Steady flow (shown in Figure 3.1)

Figure 3.1 (p. 95) (a) Flow in the x-y plane. (b) flow in terms of streamline and normal coordinates.

Streamlines • Velocity vector is tangent to the path of flow • Lines that are tangent to the velocity vectors throughout the flow field are called streamlines • Equation for a streamline:

Force balance on a Streamline

Figure 3.3 (p. 97) Free-body diagram of a fluid particle for which the important forces are those due to pressure and gravity. • The physical interpretation is that a change in fluid particle speed is accomplished by the appropriate combination of pressure gradient and particle weight along the streamline.